Fekete-like polynomials

In 2001, Borwein, Choi, and Yazdani looked at an extremal property of a class of polynomial with ±1 coefficients. Their key result was: Theorem. (See Borwein, Choi, Yazdani, 2001.) Let  f(z)=±z±z2±⋯±zN−1f(z)=±z±z2±⋯±zN−1, and ζ a primitive Nth root of unity. If N is an odd positive integer thenmaxi|f(ζi)|⩾Nwith equality if and only if N is an odd prime.Moreover, if equality holds, they gave an explicit construction for f(z)f(z). In this paper, we look at the case when N is even. In particular, we investigate the following Conjecture. Let f(z)f(z) and ζ   be as above. If N>2N>2 is an even positive integer thenmaxi|f(ζi)|⩾N+1 with equality if and only if N+1N+1 is a power of an odd prime.This conjecture was made after extensive computations. Partial results towards proving this conjecture are given.